Using the recently obtained exact results for the expectation values of operators in the sine- and sinh-Gordon models [A. B. Zamolodchikov and S. Lukyanov, Nucl. Phys. B 493, 571 (1997); V. Fateev, S. Lukyanov, A. B. Zamolodchikov, and Al. B. Zamolodchikov, Phys. Lett. B 406, 83 (1997)] we calculate the specific heat of the corresponding two-dimensional Euclidean (classical) models. We show that the temperature dependence of the specific heat of the sine-Gordon model, in the commensurate phase, has a maximum well below the Kosterlitz-Thouless transition and that the sinh-Gordon model is thermodynamically unstable in the strong-coupling regime. We give also the temperature dependence of the specific heat in the incommensurate phase of the sine-Gordon model.
I study the Sine-Gordon (SG) and Sinh-Gordon (ShG) quantum field theories with a recently introduced variational method, the relativistic continuous matrix product states (RCMPS). The main advantage is to work directly in the thermodynamic limit, and without any UV regulator. The SG model is well understood and integrable, which provides a convenient benchmark for the variational method and serves as a warm-up. RCMPS approximate the ground state of the SG model arbitrary well up to the free Fermion point [coupling β=4π in equal-time quantization convention, or b=1/2 in CFT convention], where the ground energy collapses to −∞, and some renormalized ansatz would be needed. The ShG model, while integrable, is less understood and its strong coupling regime β≈1 is subject to some controversy. RCMPS also fit the ground state of the ShG model up to approximately b=1/2, after which their predictions start to deviate substantially from the "exact" results. This is more puzzling as nothing is expected to happen physically for the ShG model at that point (eg, the ground energy density does not diverge). Either the "exact" ShG results are not exact (the analytic continuation of the SG Bethe Ansatz solution is unwarranted), or, more likely, the physical structure of the ShG ground state changes in such a way that it becomes out of reach of the RCMPS manifold for reasonable bond dimensions.